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Optical Pumping and the Hyperfine Structure of Rubidium 87

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Optical Pumping and the Hyperfine Structure of Rubidium 87 Optical Pumping and the Hyperfine Structure of Rubidium 87 Laboratory Manual for Physics 3081 Thomas Dumitrescu∗, Solomon Endlich† May 2007 Abstract In this experiment you will learn about the powerful experimental technique of optical pumping, and apply it to investigate the hyperfine structure of Rubidium in an applied external magnetic field. The main goal of the experiment is to measure the hyperfine splitting of the ground state of Rubidium 87 at zero external magnetic field. Note: In this lab manual, Gaussian units are used throughout. See [4] for the relation between Gaussian and SI units. ∗email: [email protected] †email: [email protected] 1 1 Hyperfine Structure of Rubidium In this experiment you will study the optical pumping of Rubidium in an external magnetic field to determine its hyperfine structure. In the following, we give a brief outline of the hyperfine structure of Rubidium and its Zeeman splitting in an external magnetic field. The main goal of this section is to motivate the Breit-Rabi formula, which gives the Zeeman splitting of the Hyperfine levels. For a thorough and enlightening discussion of these topics we refer to the excellent article by Benumof [1]. The ground state of alkali atoms - of which Rubidium is one - consists of a number of closed shells and one s-state valence electron in the next shell. Recall that the total angular momentum of a closed atomic shell is zero, so the total angular momentum of the Rubidium atom is the sum of three angular momenta: the orbital and spin angular momenta L and S of the valence electron, and the nuclear angular momentum I. Since Rubidium has a fairly large number of electrons and the closed shells are spherically sym- metric, we can treat the valence electron in mean field theory: the closed shells just give an effective screening contribution to the spherically sym- metric Coulomb potential of the nucleus and can otherwise be neglected. Thus the only players in our discussion will be the nucleus and the valence electron. To determine the hyperfine structure of this effective one-electron problem we have to solve the Shr¨odinger equation (H0 + Hhf )ψ = Eψ where H0 is the effective Hamiltonian describing the kinetic energy and Coulomb interaction of the electron, and Hhf is the hyperfine Hamiltonian H = ηI J µ B µ B hf · − J · − I · Here J = L+S is the total electron angular momentum, η is a proportionality constant and µJ , µI are the electron and nuclear magnetic moments respec- tively. Note that since were are only investigating the hyperfine structure of the s-wave ground state, we can neglect spin-orbit coupling. The magnetic moments are written in the form e e µ = g J and µ = g I J − J 2mc I I 2mc 2 where m is the mass of the electron, e is the charge of the electron, c is − the speed of light, and gJ , gI are the electron and nuclear gyromagnetic ratios. Note that in terms of orders of magnitude g m g 1 g , I ∼ M J ∼ 1836 J where M is the mass of a nucleon. One of the goals of this experiment will be to measure gI . It will also be convenient to introduce the total angular momentum F = J + I of the entire atom. In the ground state, the allowed values of F are just F = I S = I 1/2. The solution of ± ± the full quantum mechanical problem is not difficult - it is essentially a first-order time-independent perturbation theory problem - but somewhat lengthy. A careful derivation is given in [1]; for further reading, see also [2]. At non-zero external magnetic field, the degeneracy of the hyperfine structure is completely broken (this is the Zeeman effect) and each allowed value of m = F, F + 1, ..., F 1, F corresponds to a different energy F − − − eigenvalue; here mF is the component of F parallel to B. Relative to a convenient reference level, the hyperfine energies are given by the celebrated Breit-Rabi formula, which is usually written in terms of angular frequencies rather than energies: 1/2 F =I 1/2 ∆hf 4mF x 2 ω ± = µ g m B 1 + + x mF − B I F ± 2 2I + 1 ! " In this formula µB = e!/2mc = 1.40MHz/Gauss is the Bohr magneton, B = B , ∆ is the magnitude of the hyperfine splitting of the F = I 1/2 | | hf − and F = I + 1/2 levels at zero external magnetic field (this is another thing we want to measure in this experiment), and (g + g )µ B x = J I B ∆hf This formula depends on the sign conventions we chose for the gyromagnetic ratios; in our conventions, both gJ and gI are positive. As an aside, it is 2 1 interesting to point out that ∆hf = η! (I+ 2 ) depends only on the coefficient of I J in the hyperfine Hamiltonian, as is expected for the hyperfine splitting · at zero external field. In this experiment we will be dealing exclusively with Rubidium 87, which has I = 3/2 and consequently has F = 1 or F = 2. The hyperfine splitting between these two levels at zero external field is given by ∆hf = 6834.7MHz. The nuclear gyromagnetic ratio of Rubidium 87 is gI = 0.000999. Although the purpose of this experiment is to measure ∆hf and gI , it will be useful to plug the literature values into the Breit-Rabi formula, so that you can calculate the expected hyperfine transition frequencies. For Rubidium 87, the frequency in MHz of a hyperfine transition between a state 3 with m and a state with m 1 for F = I 1/2 is given by F F − ± 1/2 1/2 ν = 3417.34 1 + m x + x2 1 + (m 1)x + x2 0.0013978B F − F − ∓ #$ % $ % & where x = 9.2302 104 B · and in both of these formulas B is measured in Gauss. For the given exper- imental setup, it is also true that B 19.53I , where again B is measured $ el in Gauss and Iel is current generating the magnetic field measured in A. Note that for Rubidium 87 there should be a total of 6 transitions given by the formula above: for F = 1 the possible transitions are 1 0 and ↔ 0 1, and for F = 2 the possible transitions are 2 1, 1 0, 0 1 ↔ − ↔ ↔ ↔ − and 1 2. The information given above is sufficient to calculate the − ↔ − expected hyperfine transition frequencies and to analyze the data you will obtain in order to determine ∆hf and gI ; this will be discussed further be- low. For further reading we again refer to [1] and [2]. Figure 1 shows the hyperfine structure of Rubidium 87 as a function of the external magnetic field. 2 Principles of Optical Pumping In this experiment, light form a Rubidium lamp is used to optically pump Rubidium 87 vapor. In the following, we briefly outline the mechanism of optical pumping, and explain how it can be used to measure hyperfine tran- sition frequencies. As discussed further below, the most important part of the experimental setup consists of a linear arrangement of the Rubidium lamp, an optical filter and polarizer, the Rubidium vapor bulb and an op- tical detector. The light from the Rubidium lamp is filtered to only allow photons of wavelength 7947.6 A˚ to pass through the Rubidium vapor bulb. These photons are exactly of the right energy to cause a transition of the 2 2 valence electron from its S1/2 ground state to the P1/2 first excited state (see Figure 1). These filtered photons are then passed through a polarizer (a quarter wave plate) to convert them into photons of right circular polar- ization (or positive helicity); we refer to such photons as σ+ photons. If the external magnetic field is parallel to their direction of propagation, these σ+ photons can affect 2S 2 P transitions according to the following 1/2 → 1/2 dipole selection rules (these give the dominant transition processes): 4 Figure 1: Hyperfine Structure of the Low-Lying States of Rb87 (Source: [1]) ∆L = 1 ∆J = 0, 1 ∆F = 0, 1 ∆m = +1 ± ± ± F + Note that σ photons can only cause ∆mF = +1 transitions. Likewise, pho- tons of negative helicity - so-called σ photons - can only cause ∆m = 1 − F − transitions, with otherwise identical selection rules. So when a σ+ photon 2 hits a Rubidium 87 atom, it not only excites the electron to the P1/2 level, but also raises its mF by one unit. Of course, the excited state is unsta- ble and spontaneously decays back to the ground state by emitting another photon. However, the spontaneously emitted photon has arbitrary polar- ization and the decay can have ∆mF = 0, 1. Thus, on average we have + ± ∆mF = +1 for the excitation by σ photons, but ∆mF = 0 for the sponta- neous decay, giving a net average ∆mF = +1: on average, the electrons will migrate into states of higher and higher mF . This phenomenon is referred to as optical pumping. Of course m = F, F + 1, ..., F 1, F , so this F − − − process eventually has to stop. At this point the vast majority of electrons in the Rubidium 87 vapor are in the ground state of highest allowed mF (due to the close spacing of the hyperfine levels, thermal effects prevent a complete depletion of the lower-lying mF states). This process is illustrated in Figure 2. It is important to note that when the external magnetic field 5 Figure 2: The Optical Pumping Mechanism at Work (Source: Eric D.
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